{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T15:04:05Z","timestamp":1750950245856,"version":"3.37.3"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Horizon 2020,European Union","award":["952215"],"award-info":[{"award-number":["952215"]}]},{"DOI":"10.13039\/501100021856","name":"Ministero dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["Spoke 1 \u201cFutureHPC & BigData\u201d of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC)"],"award-info":[{"award-number":["Spoke 1 \u201cFutureHPC & BigData\u201d of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC)"]}],"id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Mach. Intell."],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the problems of weighted constrained sampling and weighted model counting, where we are given a propositional formula and a weight for each world. The first problem consists of sampling worlds with a probability proportional to their weight given that the formula is satisfied. The latter is the problem of computing the sum of the weights of the models of the formula. Both have applications in many fields such as probabilistic reasoning, graphical models, statistical physics, statistics, and hardware verification. In this article, we propose quantum weighted constrained sampling (QWCS) and quantum weighted model counting (QWMC), two quantum algorithms for performing weighted constrained sampling and weighted model counting, respectively. Both are based on the quantum search\/quantum model counting algorithms that are modified to take into account the weights. In the black box model of computation, where we can only query an oracle for evaluating the Boolean function given an assignment, QWCS requires <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(2^{\\frac{n}{2}}+1\/\\sqrt{\\text {WMC}})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mfrac>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mfrac>\n                    <\/mml:msup>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mtext>WMC<\/mml:mtext>\n                    <\/mml:msqrt>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> oracle calls, where <jats:italic>n<\/jats:italic> is the number of Boolean variables and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\text {WMC}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mtext>WMC<\/mml:mtext>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the normalized between 0 and 1 weighted model count of the formula, while a classical algorithm has a complexity of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega (1\/\\text {WMC})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mtext>WMC<\/mml:mtext>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. QWMC takes <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Theta (2^{\\frac{n}{2}})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0398<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mfrac>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mfrac>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> oracle calss, while classically the best complexity is <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Theta (2^n)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0398<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, thus achieving a quadratic speedup.<\/jats:p>","DOI":"10.1007\/s42484-024-00209-5","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T09:02:29Z","timestamp":1730797349000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum algorithms for weighted constrained sampling and weighted model counting"],"prefix":"10.1007","volume":"6","author":[{"given":"Fabrizio","family":"Riguzzi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,5]]},"reference":[{"key":"209_CR1","doi-asserted-by":"crossref","unstructured":"Aharonov D (1999) Quantum computation. In: Annual Reviews of Computational Physics VI. World Scientific, p 259\u2013346","DOI":"10.1142\/9789812815569_0007"},{"key":"209_CR2","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1613\/jair.2648","volume":"34","author":"F Bacchus","year":"2009","unstructured":"Bacchus F, Dalmao S, Pitassi T (2009) Solving# sat and Bayesian inference with backtracking search. J Artif Intell Res 34:391\u2013442","journal-title":"J Artif Intell Res"},{"key":"209_CR3","doi-asserted-by":"publisher","unstructured":"Bian Z, Chudak F, Macready W et al (2020) Solving SAT (and MaxSAT) with a quantum annealer: Foundations, encodings, and preliminary results. Inf Comput 275:104609. https:\/\/doi.org\/10.1016\/j.ic.2020.104609, https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0890540120300973","DOI":"10.1016\/j.ic.2020.104609"},{"issue":"1\u20132","key":"209_CR4","first-page":"1","volume":"11","author":"HL Bodlaender","year":"1993","unstructured":"Bodlaender HL et al (1993) A tourist guide through treewidth. Acta Cybernet 11(1\u20132):1\u201321","journal-title":"Acta Cybernet"},{"issue":"4\u20135","key":"209_CR5","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/(SICI)1521-3978(199806)46:4\/5<493::AID-PROP493>3.0.CO;2-P","volume":"46","author":"M Boyer","year":"1998","unstructured":"Boyer M, Brassard G, H\u00f8yer P et al (1998) Tight bounds on quantum searching. Fortschritte der Physik: Progress of Physics 46(4\u20135):493\u2013505","journal-title":"Fortschritte der Physik: Progress of Physics"},{"key":"209_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"820","DOI":"10.1007\/BFb0055105","volume-title":"25th International Colloquium on Automata, Languages and Programming (ICALP 1998)","author":"G Brassard","year":"1998","unstructured":"Brassard G, H\u00f8yer P, Tapp A (1998) Quantum counting. In: Larsen KG, Skyum S, Winskel G (eds) 25th International Colloquium on Automata, Languages and Programming (ICALP 1998), vol 1443. Lecture Notes in Computer Science. Springer, pp 820\u2013831"},{"key":"209_CR7","doi-asserted-by":"crossref","unstructured":"Chakraborty S, Fremont DJ, Meel KS, et\u00a0al (2014) Distribution-aware sampling and weighted model counting for sat. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence. AAAI Press, AAAI\u201914, pp 1722\u20131730","DOI":"10.1609\/aaai.v28i1.8990"},{"key":"209_CR8","unstructured":"Chakraborty S, Fried D, Meel KS, et\u00a0al (2015) From weighted to unweighted model counting. In: Yang Q, Wooldridge MJ (eds) 24th International Joint Conference on Artificial Intelligence (IJCAI 2015). AAAI Press, pp 689\u2013695, http:\/\/ijcai.org\/Abstract\/15\/103"},{"key":"#cr-split#-209_CR9.1","unstructured":"Chakraborty S, Meel KS, Vardi MY (2016) Algorithmic improvements in approximate counting for probabilistic inference: from linear to logarithmic SAT calls. In: Kambhampati S"},{"key":"#cr-split#-209_CR9.2","unstructured":"(ed) 25th International Joint Conference on Artificial Intelligence (IJCAI 2016). AAAI Press\/IJCAI, pp 3569-3576"},{"key":"209_CR10","doi-asserted-by":"crossref","unstructured":"Chakraborty S, Meel KS, Vardi MY (2021) Approximate model counting. In: Handbook of Satisfiability. IOS Press, p 1015\u20131045","DOI":"10.3233\/FAIA201010"},{"issue":"6\u20137","key":"209_CR11","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1016\/j.artint.2007.11.002","volume":"172","author":"M Chavira","year":"2008","unstructured":"Chavira M, Darwiche A (2008) On probabilistic inference by weighted model counting. Artif Intell 172(6\u20137):772\u2013799","journal-title":"Artif Intell"},{"issue":"1969","key":"209_CR12","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1098\/rspa.1998.0164","volume":"454","author":"R Cleve","year":"1998","unstructured":"Cleve R, Ekert A, Macchiavello C et al (1998) Quantum algorithms revisited. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 454(1969):339\u2013354","journal-title":"Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences"},{"key":"209_CR13","unstructured":"Coppersmith D (2002) An approximate Fourier transform useful in quantum factoring. ArXiv preprint quant-ph\/0201067"},{"issue":"1\u20132","key":"209_CR14","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/S0004-3702(00)00069-2","volume":"126","author":"A Darwiche","year":"2001","unstructured":"Darwiche A (2001) Recursive conditioning. Artif Intell 126(1\u20132):5\u201341","journal-title":"Artif Intell"},{"key":"209_CR15","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1613\/jair.989","volume":"17","author":"A Darwiche","year":"2002","unstructured":"Darwiche A, Marquis P (2002) A knowledge compilation map. J Artif Intell Res 17:229\u2013264. https:\/\/doi.org\/10.1613\/jair.989","journal-title":"J Artif Intell Res"},{"issue":"1\u20132","key":"209_CR16","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/S0004-3702(99)00059-4","volume":"113","author":"R Dechter","year":"1999","unstructured":"Dechter R (1999) Bucket elimination: a unifying framework for reasoning. Artif Intell 113(1\u20132):41\u201385","journal-title":"Artif Intell"},{"key":"209_CR17","doi-asserted-by":"crossref","unstructured":"Eiter T, Kiesel R (2021) On the complexity of sum-of-products problems over semirings. In: 35th AAAI Conference on Artificial Intelligence, AAAI 2021. AAAI Press, pp 6304\u20136311","DOI":"10.1609\/aaai.v35i7.16783"},{"key":"209_CR18","unstructured":"Feller W (1968) An introduction to probability theory and its applications: volume I. Wiley series in probability and mathematical statistics, John Wiley & sons"},{"key":"209_CR19","doi-asserted-by":"crossref","unstructured":"Feynman RP (1982) Simulating physics with computers. International Journal of Theoretical Physics 21(6\/7)","DOI":"10.1007\/BF02650179"},{"key":"209_CR20","doi-asserted-by":"publisher","unstructured":"Fichte JK, Hecher M, Woltran S, et\u00a0al (2018) Weighted model counting on the GPU by exploiting small treewidth. In: Azar Y, Bast H, Herman G (eds) 26th Annual European Symposium on Algorithms (ESA 2018). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 28:1\u201328:16,https:\/\/doi.org\/10.4230\/LIPIcs.ESA.2018.28","DOI":"10.4230\/LIPIcs.ESA.2018.28"},{"key":"209_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3459080","volume":"26","author":"JK Fichte","year":"2021","unstructured":"Fichte JK, Hecher M, Hamiti F (2021) The model counting competition 2020. Journal of Experimental Algorithmics (JEA) 26:1\u201326","journal-title":"Journal of Experimental Algorithmics (JEA)"},{"key":"209_CR22","unstructured":"Friesen AL, Domingos PM (2016) The sum-product theorem: a foundation for learning tractable models. In: Balcan M, Weinberger KQ (eds) 33nd International Conference on Machine Learning, ICML 2016, JMLR Workshop and Conference Proceedings, vol\u00a048. JMLR.org, pp 1909\u20131918"},{"key":"209_CR23","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1613\/jair.1.12370","volume":"73","author":"R Ganian","year":"2022","unstructured":"Ganian R, Kim EJ, Slivovsky F et al (2022) Sum-of-products with default values: algorithms and complexity results. J Artif Intell Res 73:535\u2013552","journal-title":"J Artif Intell Res"},{"key":"209_CR24","first-page":"633","volume-title":"Handbook of satisfiability","author":"CP Gomes","year":"2009","unstructured":"Gomes CP, Sabharwal A, Selman B (2009) Model counting. In: Biere A, Heule M, van Maaren H et al (eds) Handbook of satisfiability, vol 185. IOS Press, pp 633\u2013654"},{"issue":"17","key":"209_CR25","doi-asserted-by":"publisher","first-page":"3228","DOI":"10.1103\/PhysRevLett.76.3228","volume":"76","author":"RB Griffiths","year":"1996","unstructured":"Griffiths RB, Niu CS (1996) Semiclassical Fourier transform for quantum computation. Phys Rev Lett 76(17):3228","journal-title":"Phys Rev Lett"},{"key":"209_CR26","first-page":"212","volume-title":"28th annual ACM symposium on theory of computing (STOC 1996)","author":"LK Grover","year":"1996","unstructured":"Grover LK (1996) A fast quantum mechanical algorithm for database search. 28th annual ACM symposium on theory of computing (STOC 1996). ACM Press, New York, NY, USA, pp 212\u2013219"},{"key":"209_CR27","doi-asserted-by":"crossref","unstructured":"Grover LK (1996b) A fast quantum mechanical algorithm for database search. ArXiv preprint quant-ph\/9605043","DOI":"10.1145\/237814.237866"},{"issue":"2","key":"209_CR28","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1103\/PhysRevLett.79.325","volume":"79","author":"LK Grover","year":"1997","unstructured":"Grover LK (1997) Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 79(2):325","journal-title":"Phys Rev Lett"},{"key":"209_CR29","unstructured":"Gupta SS, Nagel K (1967) On selection and ranking procedures and order statistics from the multinomial distribution. Sankhy\u0101: The Indian Journal of Statistics, Series B (1960-2002) 29(1\/2):1\u201334. http:\/\/www.jstor.org\/stable\/25051587"},{"key":"209_CR30","doi-asserted-by":"publisher","first-page":"15050","DOI":"10.1103\/PhysRevLett.103.150502","volume":"103","author":"AW Harrow","year":"2009","unstructured":"Harrow AW, Hassidim A, Lloyd S (2009) Quantum algorithm for linear systems of equations. Phys Rev Lett 103:15050. https:\/\/doi.org\/10.1103\/PhysRevLett.103.150502","journal-title":"Phys Rev Lett"},{"key":"209_CR31","volume-title":"Quantum computing","author":"M Hirvensalo","year":"2013","unstructured":"Hirvensalo M (2013) Quantum computing. Springer, Natural Computing Series"},{"key":"209_CR32","unstructured":"Huang J, Chavira M, Darwiche A (2006) Solving MAP exactly by searching on compiled arithmetic circuits. In: Proceedings of the 21st National Conference on Artificial Intelligence - Volume 2. AAAI Press, AAAI\u201906, pp 1143\u20131148"},{"key":"209_CR33","unstructured":"Jerrum M, Sinclair A (1996) The Markov chain Monte Carlo method: an approach to approximate counting and integration. In: Hochbaum D (ed) Approximation algorithms for NP-hard problems. PWS Publishing Company, p 482- 520"},{"key":"209_CR34","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1613\/jair.4794","volume":"53","author":"J Kwisthout","year":"2015","unstructured":"Kwisthout J (2015) Tree-width and the computational complexity of MAP approximations in Bayesian networks. J Artif Intell Res 53:699\u2013720. https:\/\/doi.org\/10.1613\/jair.4794","journal-title":"J Artif Intell Res"},{"key":"209_CR35","doi-asserted-by":"crossref","unstructured":"Lagniez JM, Marquis P (2017) An improved decision-DNNF compiler. In: 26th International Joint Conference on Artificial Intelligence (IJCAI 2017), pp 667\u2013673","DOI":"10.24963\/ijcai.2017\/93"},{"issue":"2","key":"209_CR36","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1111\/j.2517-6161.1988.tb01721.x","volume":"50","author":"SL Lauritzen","year":"1988","unstructured":"Lauritzen SL, Spiegelhalter DJ (1988) Local computations with probabilities on graphical structures and their application to expert systems. J Roy Stat Soc: Ser B (Methodol) 50(2):157\u2013194","journal-title":"J Roy Stat Soc: Ser B (Methodol)"},{"key":"209_CR37","doi-asserted-by":"crossref","unstructured":"Madras N, Piccioni M (1999) Importance sampling for families of distributions. The Annals of Applied Probability 9(4):1202\u20131225. http:\/\/www.jstor.org\/stable\/2667147","DOI":"10.1214\/aoap\/1029962870"},{"key":"209_CR38","unstructured":"Maene J, Derkinderen V, De\u00a0Raedt L (2024) On the hardness of probabilistic neurosymbolic learning. In: Salakhutdinov R, Kolter Z, Heller K, et\u00a0al (eds) Proceedings of the 41st International Conference on Machine Learning, Proceedings of Machine Learning Research, vol 235. PMLR, pp 34203\u201334218, https:\/\/proceedings.mlr.press\/v235\/maene24a.html"},{"key":"209_CR39","doi-asserted-by":"publisher","unstructured":"Marra G, Duman\u010di\u0107 S, Manhaeve R et al (2024) From statistical relational to neurosymbolic artificial intelligence: a survey. Artif Intell 328:104062. https:\/\/doi.org\/10.1016\/j.artint.2023.104062, https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0004370223002084","DOI":"10.1016\/j.artint.2023.104062"},{"key":"#cr-split#-209_CR40.1","unstructured":"Meel K, Vardi M, Chakraborty S, et\u00a0al (2016) Constrained sampling and counting: universal hashing meets SAT solving. In: Darwiche A"},{"key":"#cr-split#-209_CR40.2","unstructured":"(ed) Beyond NP, Papers from the 2016 AAAI Workshop, Phoenix, Arizona, USA, February 12, 2016, AAAI Technical Report, vol WS-16-05. AAAI Press, http:\/\/www.aaai.org\/ocs\/index.php\/WS\/AAAIW16\/paper\/view\/12618"},{"key":"209_CR41","unstructured":"Mosca M (1999) Quantum computer algorithms. PhD thesis, University of Oxford. 1999"},{"issue":"2","key":"209_CR42","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s10773-019-04354-7","volume":"59","author":"K Nagata","year":"2020","unstructured":"Nagata K, Nakamura T (2020) Some theoretically organized algorithm for quantum computers. Int J Theor Phys 59(2):611\u2013621","journal-title":"Int J Theor Phys"},{"key":"209_CR43","first-page":"1","volume-title":"Mathematical digital quantum computation by means of much more logical skills","author":"K Nagata","year":"2024","unstructured":"Nagata K, Nakamura T (2024) Mathematical digital quantum computation by means of much more logical skills. Mathematics and Foundations, Quantum Studies, pp 1\u201315"},{"key":"209_CR44","doi-asserted-by":"publisher","DOI":"10.1088\/978-0-7503-4700-6","volume-title":"Simplified quantum computing with applications","author":"K Nagata","year":"2022","unstructured":"Nagata K, Diep DN, Farouk A et al (2022) Simplified quantum computing with applications. IOP Publishing"},{"key":"209_CR45","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1007\/s10773-020-04661-4","volume":"60","author":"T Nakamura","year":"2021","unstructured":"Nakamura T, Nagata K (2021) Physics\u2019 evolution toward computing. Int J Theor Phys 60:70\u201379","journal-title":"Int J Theor Phys"},{"key":"209_CR46","unstructured":"Naveh Y, Rimon M, Jaeger I, et\u00a0al (2006) Constraint-based random stimuli generation for hardware verification. In: Proceedings of the 18th Conference on Innovative Applications of Artificial Intelligence - Volume 2. AAAI Press, IAAI\u201906, p 1720-1727"},{"key":"209_CR47","volume-title":"Quantum computation and quantum information: 10th","author":"M Nielsen","year":"2010","unstructured":"Nielsen M, Chuang I (2010) Quantum computation and quantum information: 10th, Anniversary. Cambridge University Press","edition":"Anniversary"},{"key":"209_CR48","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1613\/jair.1236","volume":"21","author":"JD Park","year":"2004","unstructured":"Park JD, Darwiche A (2004) Complexity results and approximation strategies for MAP explanations. J Artif Intell Res 21:101\u2013133. https:\/\/doi.org\/10.1613\/jair.1236","journal-title":"J Artif Intell Res"},{"key":"209_CR49","doi-asserted-by":"crossref","unstructured":"Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann","DOI":"10.1016\/B978-0-08-051489-5.50008-4"},{"key":"209_CR50","doi-asserted-by":"publisher","unstructured":"Riguzzi F (2013) MCINTYRE: a Monte Carlo system for probabilistic logic programming. Fund Inform 124(4):521\u201354. https:\/\/doi.org\/10.3233\/FI-2013-847","DOI":"10.3233\/FI-2013-847"},{"key":"209_CR51","doi-asserted-by":"publisher","unstructured":"Riguzzi F (2020) Quantum weighted model counting. In: De Giacomo G, Catala A, Dilkina B, et\u00a0al (eds) 24th European Conference on Artificial Intelligence (ECAI 2020). IOS Press, Amsterdam, Berlin, Washington DC, pp 2640\u20132647,https:\/\/doi.org\/10.3233\/FAIA200401","DOI":"10.3233\/FAIA200401"},{"key":"209_CR52","doi-asserted-by":"crossref","unstructured":"Riguzzi F (2022) Foundations of probabilistic logic programming languages, semantics, inference and learning, second edition. River Publishers","DOI":"10.1201\/9781003338192"},{"key":"209_CR53","unstructured":"Sang T, Beame P, Kautz HA (2005) Performing Bayesian inference by weighted model counting. 20th National Conference on Artificial Intelligence (AAAI 2005). AAAI Press, Palo Alto, California USA, pp 475\u2013482"},{"key":"209_CR54","unstructured":"Sang T, Beame P, Kautz HA (2007) A dynamic approach for MPE and weighted MAX-SAT. In: 20th International Joint Conference on Artificial Intelligence (IJCAI 2007). AAAI Press\/IJCAI, pp 173\u2013179"},{"key":"209_CR55","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/B978-0-444-88650-7.50019-6","volume-title":"4th Conference Conference on Uncertainty in Artificial Intelligence (UAI 1988)","author":"PP Shenoy","year":"1990","unstructured":"Shenoy PP, Shafer G (1990) Axioms for probability and belief-function proagation. In: Shachter RD, Levitt TS, Kanal LN et al (eds) 4th Conference Conference on Uncertainty in Artificial Intelligence (UAI 1988). North-Holland, pp 169\u2013198"},{"key":"209_CR56","doi-asserted-by":"crossref","unstructured":"Shor PW (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: 35th Annual Symposium on Foundations of Computer Science (FOCS 1994). IEEE Press, pp 124\u2013134","DOI":"10.1109\/SFCS.1994.365700"},{"key":"209_CR57","first-page":"670","volume-title":"19th National Conference on Artificial Intelligence, AAAI\u201904","author":"W Wei","year":"2004","unstructured":"Wei W, Erenrich J, Selman B (2004) Towards efficient sampling: exploiting random walk strategies. In: McGuinness DL, Ferguson G (eds) 19th National Conference on Artificial Intelligence, AAAI\u201904. AAAI Press \/ The MIT Press, San Jose, California, USA, pp 670\u2013676"},{"key":"209_CR58","doi-asserted-by":"publisher","unstructured":"Ying M (2010) Quantum computation, quantum theory and AI. Artif Intell 174(2):162\u2013176. https:\/\/doi.org\/10.1016\/j.artint.2009.11.009, special Review Issue","DOI":"10.1016\/j.artint.2009.11.009"},{"key":"209_CR59","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1613\/jair.305","volume":"5","author":"NL Zhang","year":"1996","unstructured":"Zhang NL, Poole DL (1996) Exploiting causal independence in Bayesian network inference. J Artif Intell Res 5:301\u2013328","journal-title":"J Artif Intell Res"}],"container-title":["Quantum Machine Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-024-00209-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42484-024-00209-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42484-024-00209-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T16:12:17Z","timestamp":1734970337000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42484-024-00209-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,5]]},"references-count":61,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["209"],"URL":"https:\/\/doi.org\/10.1007\/s42484-024-00209-5","relation":{},"ISSN":["2524-4906","2524-4914"],"issn-type":[{"type":"print","value":"2524-4906"},{"type":"electronic","value":"2524-4914"}],"subject":[],"published":{"date-parts":[[2024,11,5]]},"assertion":[{"value":"4 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"73"}}